System and method of obtaining entrained cylindrical fluid flow

ABSTRACT

A method and system for entraining fluids is provided. The method and system may be used to create filaments. As one example, a filament may be produced by entraining a first fluid within a second fluid by flowing a third fluid, the flowing third fluid at least partly constraining the second fluid in at least one dimension. As another example, a filament may be produced by entraining a first fluid within a second fluid based on a model of a dynamic response of the first and second fluids as functions of densities of the first and second fluids, viscosities of the first and second fluids, Reynolds numbers of the first and second fluids, and Weber numbers of the first and second fluids.

REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application No. 60/648,102, filed Jan. 27, 2005. U.S. Provisional Application No. 60/648,102, filed Jan. 27, 2005 is hereby incorporated by reference herein in its entirety.

FIELD

The present application relates to a method and apparatus for obtaining entrained fluid flow and more specifically, a method and apparatus for obtaining a stable entrained cylindrical fluid flow.

BACKGROUND

Formation of bubbles or filaments in a fluid, such as a liquid or gas, may be of interest. For example, bubbles may be used in a variety of scientific fields, such as in the fields of biomedicine (e.g., in diagnosis, as potential gene therapy vectors, to convey tiny amounts of therapeutic gases in the bloodstream without the risk of embolism, etc.), advanced physics studies (bubble sonoluminiscence, damping agents in neutron spallation sources, etc.), chemical engineering, and as a strong allied in environmental protection (e.g., microbubble drag reduction in marine transport, dissolved air flotation water depuration techniques, etc.). As another example, filaments may be used to create fibers or the like.

Forming the bubbles or filaments in the fluid may be difficult. The interface between the bubbles or filaments and the surrounding fluid may be complicated, thereby making it difficult to control a system to make stable and acceptable bubbles or filaments. Therefore, there is need for a system that enables the creation of stable and acceptable bubbles or filaments.

BRIEF SUMMARY

In one aspect of the invention, a filament is produced by entraining a first fluid within a second fluid by flowing a third fluid, the flowing third fluid at least partly constraining the second fluid in at least one dimension. The third fluid may flow through a variety of methods, such as by applying pressure to the fluid flow. The third fluid flow may constrain the second fluid axisymmetrically. Further, the third fluid may apply a force on the second fluid to at least partly drag or withdraw the second fluid from its nozzle and/or reservoir. In turn, the second fluid may apply a force on the first fluid to at least partly drag or withdraw the first fluid from its nozzle and/or reservoir. In this manner, the force used to entrain the first fluid may be more due to dragging or withdrawing rather than on a direct force on the first fluid (such as increasing pressure applied to the reservoir housing the first fluid). Specifically, if the force used to entrain the first fluid may be due in part, or entirely, on the fluid properties of the first and second liquid. For example, the first fluid may be entrained within the second fluid based on the viscosities of the first and second fluids. When a filament is produced, a part of the second fluid and a part of the first fluid (such as if the first fluid is a liquid) may be solidified in order to form a fiber. When the first fluid is withdrawn, in order to maintain a constant pressure, the first fluid's reservoir may need to be filled with additional first fluid. Alternatively, instead of applying no pressure on one or more of the fluids (such as moving the fluid only by dragging or withdrawal), one or more of the fluids may be “sucked upward,” so that an additional amount of a dragging force is required to move the fluid in the direction opposite to the force sucking upward.

In another aspect of the invention, a filament being produced by entraining a first fluid within a second fluid based on a model of a dynamic response of the first and second fluids as functions of densities of the first and second fluids, viscosities of the first and second fluids, Reynolds numbers of the first and second fluids, and Weber numbers of the first and second fluids. The model of the dynamic response may define a phase space of a stable entraining of the first fluid within the second fluid. Further, the system may be controlled such that values of the Reynolds numbers of the first and second fluid and Weber numbers of the first and second fluid may be selected to be in the phase space of the stable entraining of the first fluid within the second fluid. The model may be derived numerically and/or analytically. When a filament is produced, a part of the second fluid and a part of the first fluid (such as if the first fluid is a liquid) may be solidified in order to form a fiber.

The foregoing summary has been provided only by way of introduction. Nothing in this section should be taken as a limitation on the following claims, which define the scope of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an example of a block diagram of the system for controlling fluid flow.

FIG. 2 is an example of a partial cross-section depicting interaction of two flowing fluids.

FIG. 3 is an example of a depiction of three flowing fluids.

FIG. 4 is another example of a cross-section depicting interaction of two flowing fluids.

FIG. 5 is another example of a cross-section depicting interaction of three flowing fluids.

FIG. 6 is example of a cross-section depicting interaction of two flowing fluids, with the outer fluid driven by co-electrospinning.

FIG. 7 is example of a cross-section depicting interaction of three flowing fluids, with the outermost fluid driven by co-electrospinning.

FIG. 8 is another example of a cross-section depicting interaction of three flowing fluids, with the outermost fluid driven by co-electrospinning.

FIG. 9 a depicts a graph of the jetting-dripping transition in the {β, We} plane (Re=11.2 and α=1).

FIG. 9 b depicts a graph of several jetting-dripping transition lines separating the {Re, We} plane into absolutely unstable (dripping) and convectively unstable (jetting) regions, for α=1, α=1.03 and diverse values of β.

FIG. 9 c depicts a graph of absolute/convective instability transition loci in the {Re, We} plane for different values of {α, β}.

FIG. 10 depicts the theoretical jetting-dripping transition for various (α, β, Re, We).

FIG. 11 depicts a graph of a phase diagram with horizontal axis z_(N), the nozzle location, and vertical axis dR/dz(z_(N))=sqrt (μ/4μ₀) cot θ_(N).

FIGS. 12 a-f summarize one set of experimental results.

FIG. 13 summarize another set of experimental results.

FIG. 14 depicts a graph ploting the experimental results of FIG. 13 for three representative liquids.

FIG. 15 depicts the flow leading to (after solidification downstream) creation of a single fiber containing multiple interior cylinders which are either hollow, if a gas is used as the inner fluid, or are composed of a second material.

FIG. 16 depicts nozzles aligned along a line, allowing the fluid inside the nozzle to be entrained by a two-dimensional viscous flow in a thin liquid film.

FIG. 17 illustrates a sheet containing parallel tubes prepared by aligning nozzles along a line.

FIG. 18 depicts an arrangement of parallel ribbons containing parallel long holes along them can be packed in the form of a “long brick” with many parallel long holes along their length.

DETAILED DESCRIPTION OF THE EMBODIMENTS

As discussed in the background, there are many applications for bubbles or filaments. For example, in the context of filaments, hollow cylinders or tubes and compound fibers with an inner diameter of micron and sub-micron sizes have a variety of applications. One of the most promising is in fiber optics. If glass is used as the exterior fluid and the interior is either left hollow or filled with a material with a different index of refraction, one can create optical waveguides whose properties can be varied in a flexible way by changing the composition of the two fluids and by changing the dimensions of the fiber. In other contexts, if a polymer solution or melt is used as the exterior fluid and then solidified, the resultant tubes may be used in membrane filters, capillaries for chromatography as well as bioreactors and as cylindrical microfluidic channels. Compound fibers comprised of an inner metal core and an outer insulator sheath may be used as cables in microelectronic applications. The outer sheath may also be a metal. Possible applications for a metal structure comprised of one or more tubes include: (i) heat exchangers for biology, biomedicine, and cardiovascular applications; (ii) surgical devices; (iii) scientific & laboratory optical or microfluidic instruments; and (iv) reactors for chemical reactions. Other possible materials that may be used include polymers such as polysaccharides and polypeptides.

A method and system are provided in order to create stable and/or acceptable bubbles or filaments. One way to create filaments is to produce an extensional flow, such as by entraining a fluid within another fluid. For example, fluid that is more viscous than the fluid being entrained may flow from a nozzle. The combination of the entrained fluid and the outer fluid may be used to create hollow cylinders (for example, by entraining a gas within a fluid) or compound fibers on micron and sub-micron scales. As discussed herein, fluid may comprise a liquid, a gas, or a combination of a liquid and a gas.

As discussed in the background, the ability to create stable and/or acceptable bubbles or filaments depends, at least in part, on the forces at the interface between the multiple fluids. As discussed below, a model is created to analyze the interface between fluids. The model may be used in order to adjust the system (including dynamically adjusting the system) in order to create bubbles or filaments. For example, creating a filament by entraining a fluid within another fluid may be achieved by applying a force to the fluids (including the inner fluid or the outer fluid). The model, discussed below, may be used to determine the forces to entrain the fluid.

The force may be applied directly or indirectly to the fluids in order to produce an extensional flow. There are several examples of a direct force that may be applied to a fluid. Where the fluid is in a tank or a reservoir, pressure may be applied so that the fluid is forced from a nozzle connected to the tank or reservoir. Where the fluid is conductive, a voltage may be applied so that the electric field generated from the voltage applies a force to the conductive fluid. Co-electrospinning is an example of using a voltage to apply a force to the fluid. Further, a combination of direct forces, such as applying pressure and co-electrospinning, may be used to apply a direct force to a fluid.

One or more of the fluids may also be subject to an indirect force. As one example, the inner fluid may be entrained, not by any force applied directly to it (such as applying pressure to the inner fluid or co-electrospinning of the inner fluid), but by proximity of the inner fluid to movement of another fluid (such as movement of an outer fluid). The movement of the outer fluid may be caused by a direct force (such as by applying pressure to the outer fluid or co-electrospinning), or may be caused by an indirect force applied to it as well. In effect, the inner fluid is dragged along or withdrawn by the outer fluid, rather than pushed along.

Further, a fluid may be subject to multiple direct or indirect forces. For example, a fluid may be subject to the multiple direct forces of pressure and co-electrospinning. As another example, the fluid may be subject to an indirect dragging force (such as caused by an outer fluid dragging an inner fluid) and may be subject to a direct force (such as pressure applied to tank supplying the inner fluid). In the case of multiple direct and indirect forces, one may compare the amount of each of the forces. A fluid that has a greater indirect force may be considered to have a greater dragging force than pushing force. Similarly, a fluid that has a greater direct force may be considered to have a greater pushing force than dragging force.

Referring to FIG. 1, there is shown an example of a block diagram of the system 100 for controlling fluid flow. The system 100 may include a control system 102 that may select the parameters so that the system performs acceptably (such as entrainment of an inner fluid) and may send outputs so that the system operates under the selected parameters. The control system 102 may include a processor 104 that accesses a memory 106. The memory may include a programs or models of dispersions, as discussed in more detail below. The processor 104 may determine the parameters for the system and may control the system, via I/O 108. The control of the system may be performed in variety of ways, such as through voltage control 110 (which may control co-electrospinning), through pressure control 112 (which may control the pressure of one or more of the fluids by applying pressure to a tank housing the fluid), nozzle angle 114 (which may adjust the nozzle angle that ejects the fluid), and radius control 116 (which may control the radius of one or more of the nozzles).

A model may be derived for the interaction of the flowing fluids. Referring to FIG. 2, there is shown an example of an interaction of flowing fluids. Specifically, FIG. 2 depicts a flow-focusing setup illustrating the spatio-temporal stability of a core liquid jet produced by flow focusing (FF). The nozzle may emit a liquid filament 1 which is surrounded by and pulled by the indirect force of the co-flowing liquid 2. Co-flowing liquid 2 may be subject to a direct force of being pressurized. Further, co-flowing liquid 2 may be immiscible with liquid filament 1. Though FIG. 2 depicts an inner liquid (liquid filament 1) and an outer liquid (co-flowing liquid 2), any combination of liquids or gases may be used for the inner and outer fluid.

The liquid filament 1 and co-flowing liquid 2 may have liquid viscosities μ_(1,2) and densities ρ_(1,2) cylindrical coordinates r, z. In analyzing the spatio-temporal stability depicted in FIG. 2, one may make the following assumptions: (i) the liquid filament 1 and co-flowing liquid 2 are viscous and immiscible with an interfacial surface tension of σ; (ii) the Reynolds number of the co-flowing shell Re₂=ρ₂VDμ₂ ⁻¹ is large enough to ensure a uniform stream-wise velocity Ve_(z) around the liquid jet, e_(z) being the axial unit vector; (iii) the core liquid jet diameter d_(j)=2R_(j) is roughly uniform and much smaller than the orifice diameter D, d_(j)/D<<1, so that the liquid jet may be considered to flow in an unbounded co-flowing liquid domain moving with uniform velocity V; this simplification (both in the theory and in the setup) attempts to enforce cylindrical homogeneity. This simplification circumvents the complexity of a downstream variation of the flow regime (and hence, excludes the possibility of heterogeneous stability behavior at different axial locations). Thus purely cylindrical flow patterns are considered, and disregarding the effects of local instability sources (e.g., oscillation of the meniscus); (iv) the axial length l required for a full momentum diffusion from the shell to the core, l˜92 ₁Vμ⁻¹d² _(j), is small compared to the axial length of the orifice L, 1/L<<1, and thus both liquid streams may move with a similar uniform axial velocity V; (v) no azimuthal modes are considered (i.e., pure axisymmetric motion), which is a simplification supported by both theory and experiment, on the jetting-dripping transition.

With these assumptions, the system response may be analyzed for small perturbations proportional to e^(i(kz−Ωt)). In analyzing the dispersion relation for a viscous liquid cylinder in an immiscible viscous ambient liquid, as depicted in FIG. 2, wave frequency Ω, time t, wave number K, and coordinates {r, z} may be scaled and made dimensionless with with V/R_(j), R_(j)/V, 1/R_(j) and R_(j), respectively. Four dimensionless parameters may arise: the Reynolds and Weber numbers, and the liquid density and viscosity ratios, as shown in the following equation: Re=ρ ₁VR_(j)μ₁ ⁻¹, We=ρ₁V²R_(j)σ⁻¹, α=ρ₂/ρ₁, β=μ₂/μ₁  (1)

The conservation equations of mass and momentum of the liquid flow, together with the boundary conditions at the jet surface (including normal and tangential stress balance) and at infinity, lead to the dispersion relation between the perturbation wave frequency Ω and its wavenumber k. Thus, a dimensionless dispersion relation S({circumflex over (ω)}, x; Re, We, α,β)=0 (with {circumflex over (ω)}=R_(j)ΩV⁻¹ and x=R_(j)k) may be simplified with the aid of a mathematical tool, such as Mathematica V5.01, by introducing a speed V of the two fluids into: $\begin{matrix} {{S \equiv {{\left( {x^{2} - y_{1}^{2}} \right)\left\lbrack {\frac{N\left( {x,y_{1},y_{2},\beta} \right)}{M\left( {x,y_{1},y_{2},\beta} \right)} + {2\left( {1 - \beta} \right)}} \right\rbrack} - {\frac{{Re}^{2}}{We}\left( {1 - x^{2}} \right)}}} = 0} & (2) \end{matrix}$

where “viscous” wave numbers are defined for both liquids as: y ₁ ² =x ² −iRe{circumflex over (ω)}, y ₂ ² =x ² −iαβ ⁻¹ Re{circumflex over (ω)},  (3)

and functions N and M are expressed as: N≡2xβy ₁ y ₂ [K ₀(y ₂)I ₁(y ₁)y ₁ +I ₀(y ₁)K ₁(y ₂)y ₂ ]+x[x ²(β−1)−y ₁ ² +βy ₂ ²]² I ₀(x)I ₁(y ₁)K ₀(x)K ₁(y ₂)+4x ³ y ₁ y ₂(β−1)² I ₀(y ₁)I ₁(x)K ₀(y ₂)K ₁(x)−−y ₂ I ₁(y ₁)K ₀(y ₂){[x ⁴ +y ₁ ² y ₂ ² +x ²(y ₁ ² −y ₂ ²)]βI ₁](x)+[y ₁ ⁴ +x ⁴(1−2β)²−2x ² y ₁ ²(β−1)]I ₀(x)K ₁(x)}+y ₁ I ₀(y ₁)K ₁(y ₂){[x ⁴(β−2)²+2x ² y ₂ ²β(β−1)+β² y ₂ ⁴ ]I ₁(x)K ₀(x)+[x ²(x ² −y ₁ ²)+y ₂ ²(x ² +y ₁ ²)]βI ₀(x)K ₁(x)}  (4) M≡x{[y ₂ K ₀(y ₂)K ₁(x)−xK ₀(x)K ₁(y ₂)](y ₁ ² −x ²)I ₁(x)I ₁(y ₁)+β[y ₁ I ₀(y ₁)I ₁(x)−xI ₀(x)I ₁(y ₁)](y ₂ ² −x ²)K ₁(x)K ₁(y ₂)}  (5)

Here, I₀, I₁, K₀, and K₁ stand for the modified Bessel functions of order 0 and 1. Further, the limit α→0, β→0 yields: $\begin{matrix} \left. \frac{N}{M}\rightarrow{{\frac{\left( {x^{2} + y_{1}^{2}} \right)^{2}}{x\left( {x^{2} - y_{1}^{2}} \right)}\frac{I_{0}(x)}{I_{1}(x)}} - {\frac{4x^{2}y_{1}}{\left( {X^{2} - y_{1}^{2}} \right)}\frac{I_{0}\left( y_{1} \right)}{I_{1}\left( y_{1} \right)}}} \right. & (6) \end{matrix}$

This leads to the liquid-vacuum dispersion relation of: $\begin{matrix} {{S \equiv {{\left( {x^{2} + y_{1}^{2}} \right)^{2}\frac{I_{0}(x)}{I_{1}(x)}} - {4x^{3}y_{1}\frac{I_{0}\left( y_{1} \right)}{I_{1}\left( y_{1} \right)}} + {2{x\left( {x^{2} - y_{1}^{2}} \right)}} + {\frac{{Re}^{2}}{We}{x\left( {x^{2} - 1} \right)}}}} = 0} & (7) \end{matrix}$

One may also use a spectral numerical code developed for the stability analysis of swirling flows in pipes. Here, the linearized equations may be discretized in the r-direction using Chebyshev spectral collocation points (n_(i) points for the inner fluid and n_(e) points for the outer one). For a given wave frequency Ω, one may linearize the non-linear (quadratic) eigenvalue problem for the wave number x using a linear companion matrix method. The resulting linear eigenvalue problem may be solved numerically with the help of an eigenvalue solver subroutine (DGVCCG from the IMSL library) which provides the entire spectrum of eigenvalues and eigenfunctions. Spurious eigenvalues may be excluded by comparing the computed spectra obtained for different values of the number of collocation points. The use of the numerical procedures allows: (i) to check that both techniques render the same results, and (ii) to use the numerical spectral technique in further studies to investigate the influence of other parameters, such as the existence of other basic velocity profiles or the case of a bounded liquid flow, which are not considered in this work.

Based on the preceding, one may determine several things. First, to change the reference system from a traveling observer to a fixed observer anchored at the nozzle, one simply needs to replace the wave frequency Ω=VR_(j) ⁻¹{circumflex over (ω)} by Ω′=VR_(j) ⁻¹(ω−x) in the dispersion relation. The new ω is the dimensionless wave frequency for the static observer. This may be proved by retaining the linearized convective terms Vu_(z) and VR_(z) in the momentum equation and the kinematic condition at the interface, respectively. In a fixed coordinate system, y₁ and y₂ are: y ₁ =±[x ² −iRe(ω−x)]^(1/2) , y ₂ =±[x ² −iαβ ⁻¹ Re(ω−x)]^(1/2)  (8)

Second, both roots of the viscous wave numbers y₁ and y₂ may be explored to encompass all potential sources of spatio-temporal instability. The theoretical frame supporting the results does not require such a precaution. The dispersion model may be symmetric with respect to y₁, so that identical {(ω, x} pairs may result, regardless of which root ±y is chosen. This may not be the case with y₂, whose positive and negative roots may yield completely different results, implying different wave solution pairs {ω, x}.

Third, consistent with the spatial-temporal stability analysis, both ω and x may be considered to be complex variables. Therefore, ω=ω_(r)+iω_(i) and x=x_(r)+i x_(i), {ω_(r), ω_(i),x_(r),x_(i)} being real numbers (“oscillation frequency”, “local growth rate”, “wave number” and “spatial growth rate”, respectively). Following the well established spatial-temporal formalism to describe the absolute/convective character of axisymmetric instabilities in the {R_(e), W_(e), α, β} parametrical space of our problem, solutions of dω/dx=0 in the dispersion relation (2) with non-zero imaginary parts of ω and x. The system may be defined to be absolutely unstable if dω/dx=0 for Im(x)<0, and Im(ω)>0. One may specially care to choose all solutions whose spatial branches departing from the saddle point dω/dx=0 originate from distinct halves of the x-plane, i.e., the only ones providing the absolute instability growth rate.

A specific application of the above formulation may be applied in the low Reynolds regime (discussed with regard to FIG. 11). Specifically, an analytical solution for the formation of gas spouts in an extensional, high viscosity liquid flow that presumably would allow the formation of extremely thin and long holes in a viscous liquid bulk. The analytical solution may be relevant to the formation of microbubbles from gas ligaments in co-flowing liquids. Specifically, the solution may address whether a long gas filament in a co-flowing liquid may be stable for any combination of densities, viscosities, surface tension, and co-flow velocity. Further, the solution may address a possible transition from bubbling to jetting such as in liquid jets.

FIG. 3 is another example of a depiction of three concentric fluids. As discussed below, the fluids selected may result in a low Reynolds regime. As illustrated in FIG. 3, the entraining flow in the exterior flow may be controlled by imposing a converging flow in the fluid surrounding the two inner fluids. Analysis using the long-wavelength model for entrainment, describes how, with a suitably strong exterior entraining flow, such a compound fluid cylinder may be stabilized against breakup into droplets due to surface tension. This means that the liquid compound cylinder may be easily solidified into solid fibers with an interior core of one material and an exterior cylindrical sheath of a second material. This setup also allows air to be entrained as the inner fluid, and therefore can be used to manufacture hollow cylinders.

To create a stable entrained inner cylinder, the innermost fluid may be less viscous than the entraining fluid, the two fluids may be immiscible, and both may have finite surface tension. Given any two such fluids, if the flow rates and reservoir pressures are controlled to satisfy the continuous portion of the phase diagram of FIG. 11, the diameter of the inner fluid may be tuned to a desired value, and the inner fluid may remain a continuous cylinder of, in theory, infinite length. Referring to FIG. 11, there is shown a phase diagram with horizontal axis ZN, the nozzle location, and vertical axis dR/dz(z_(N))=sqrt(μ/4μ₀) cot θ_(N). FIG. 11, in the low Reynolds regime, depicts that the entrainment dynamics at the threshold may be a function of conditions at the nozzle, in particular, the dynamic contact angle and the nozzle location relative to the imposed entraining flow. Different nozzle conditions may result in different values for the entrainment threshold, here expressed in terms of a dimensionless quantity, Ca, the capillary number, which characterizes the ratio of viscous stress in the entraining flow versus surface tension. As shown in FIG. 11, values corresponding to entrainment transition at one particular Ca_(c) lie on a line segment, depicted as a dashed line in the figure. Also, the minimum value of the inner fluid diameter may be determined by the limits of the continuum approximation.

Either the spout or the entraining fluid or both may be an emulsion or a dispersion, if the concentration of the droplets or particles is kept sufficiently dilute to maintain Newtonian rheological properties. For example, carbon black may be incorporated into the sheath to provide antistatic properties. In contrast to existing techniques, e.g. chemical synthesis or co-electrospinning, in one aspect of the invention, the process may rely only on the fluid mechanics for the entrainment; therefore, it may be robust and may function for a wide variety of materials. For co-electrospinning, which also partially relies on a mechanical process, to be successful the fluid pair used may satisfy certain conditions on their viscosities, densities and electric conductivities. In one aspect, the process may have the interior fluid be less viscous than the exterior fluid. Alternatively, in another aspect, the process may rely exclusively on chemical synthesis or co-electrospinning, or may rely on a combination of chemical synthesis, co-electrospinning and/or fluid mechanics.

With the same configuration but working in a different range of flow rates, one can allow the compound liquid cylinder to breakup into micron and submicron-sized compound liquid droplets or bubbles. Again, because such droplets may be prepared of a size comparable to the wavelength of light, they may have unique optical properties and may have potential applications, such as in the creation of an optical Bragg switch. Such compound drops may also have applications in controlled drug delivery, catalysis, and in various cosmetic and food applications which involve emulsions.

Referring to FIG. 4, there is shown another example of flow of two fluids. Fluid 404 may be injected through tube 402. The liquid may form a meniscus 410, which may be focused by an external fluid, such as fluid 406. Fluid 406 may comprise an outermost axisymetric flow, and may be composed of a polymer. Further, an axisymetric entrance, such as an orifice or a channel 408, may be used to focus the outermost fluid.

Referring to FIG. 5, there is shown yet another example of flow of three fluids. Similar to FIG. 4, the innermost fluid 504, for purposes of this figure termed the first fluid, may be injected from tube 402 and form meniscus 410. Further, fluid 506, termed the second fluid, is injected from tube 502, which is coaxial with tube 402. A third fluid, 510, may focus the second fluid. Further, an axisymetric entrance, such as an orifice or a channel 508, may be used to focus the third fluid 510. The first, second and third fluids may comprise either a liquid or a gas. Further, fluids of different viscosities may be selected. For example, the first, second, and third fluids may be air, polymer, and air, respectively; or may be liquid or gas, polymer, and polymer. Any combination may be used.

Referring to FIGS. 6 and 7, there is shown another example of flow of two and three fluids, respectively, using an outermost fluid being focused by electrostatic forces. For example, in FIG. 6, there is shown a meniscus 602 which produces fluid flow 604. The fluid flow 604 is focused by outer fluid 606, which may be a liquid focused by electrostatic forces. If the fluid flow 604 is entirely due to the dragging force applied by outer fluid 606, then the pressure associated with the meniscus may be controlled to be constant. Specifically, as the fluid flow 604 is drawn, the meniscus may be replenished such that there is no or little pressure change. This control of the pressure may be accomplished by pressure sensors and valves associated with a tank housing the innermost fluid. FIG. 7 depicts a three fluid system, with an innermost fluid 708 flowing from meniscus 602. A second fluid 706 may also flow as depicted in FIG. 7. A third fluid 704 may comprise an outermost fluid that is focused using electrostatic forces. The third fluid 704 may thus draw the second fluid 706, which may in turn draw the first fluid 708.

FIG. 8 depicts an expanded diagram of FIG. 7, showing tubes 806, 804, and 802 for the first, second and third fluids, respectively. Further, the first, second and third fluids may enter a liquid bath. Either before or in the liquid bath, one, some or all of the fluids may be solidified. In the case of three liquids, each of the liquids may be solidified. In the case where one or more of the fluids is a liquid and the remaining fluids are a gas, the liquids may be solidified. In order to collect the solidified filaments, a take-up reel may be used, as is known in the art. The take-up reel may thus spool the fibers produced.

Analyzing the dispersion model, one may investigate the stability, and the type (absolute vs. convective) of the instabilities that lead to break-up of hollow capillary jets in a co-flowing unbounded liquid medium. This analysis may be over a wide range (such as six orders of magnitude) of both Reynolds and Weber (We) numbers. In contrast to what occurs with liquid jets, hollow jets in unbounded co-flowing liquids may be absolutely unstable, leading to local bubbling for all values of the Reynolds and Weber numbers of the co-flowing liquid. The presence of a gas (or any other fluid with finite viscosity) inside the jet may elicit a transition from the absolute to convective character of the instability for a finite {Re, We} pair. This transition may correspond to the bubbling-jetting crisis similar to the dripping-jetting transition for capillary liquid jets. Experiments analyzing previously collected data and discussed below support these conclusions, which delimit the parametrical realm of micro-bubbling in unbounded co-flowing liquids.

The dispersion model may be analyzed in several ways. The dispersion model may be easily reduced to a limit by setting β=0. At this limit, there may be a single wave solution, absolutely unstable, for both positive and negative roots of y₁ ² Traveling away from this limit solution, under fixed Re and α (e.g. Re=11.2; α=1), β may be changed to evaluate the jetting-dripping transition loci. Referring to FIG. 9 a, there is shown a graph of the jetting-dripping transition in the {β, We} plane (Re=11.2 and α=1). Two theoretical transition curves are depicted in FIG. 9 a corresponding to positive (WP) and negative (WN) y₂ ² roots are plotted. Both solutions may collapse when β→0. Of note, the WN-branch is the one to provide physically meaningful jetting-dripping loci {β,We}. In evolving on the {We,β} space in FIG. 9 a (moving from jetting towards dripping), one first crosses the negative branch WN. Clearly, as one moves deeper in the lower {Re, We} region (smaller flow rates), one eventually hits the second branch WP; however, this may be physically irrelevant since absolute instability modes are already activated (leading to dripping) once the transition through the WP-branch has taken place. The formal procedure to determine the jetting-dripping loci may be based on the simultaneous solution of the dispersion relation along with the constraint that the group velocity be zero. The resulting solutions {ω*,x*} may happen to be saddle points; and some further checks may be required before identifying them as jetting-dripping transition points. In short, one may look for points {ω*,x*} where S=0 and ∂S/∂x=0, with x*_(i)<0, ω*_(i)=0. In addition, the branches issuing from the saddle point may cross the real x axis so as to define a range of unstable wavelengths. These conditions may identify the associated parametrical point {Re*,We*,α*,β*} as being located at the boundary (i.e., ω*_(i)=0) of an absolutely unstable behavior.

FIG. 9 b depicts several jetting-dripping transition lines separating the {Re,We} plane into absolutely unstable (dripping) and convectively unstable (jetting) regions, for α=1, α=1.03 and diverse values of β. The limit of the dispersion relation (by setting β=0) is represented as a dotted line. The jetting-dripping lines may exhibit an elbow or fold-up which is not present in the limit (β=0). Thus, dripping may be obtained by sufficiently depressing either Re or We. The plot also shows the limit to closely fit the dispersion model when α=0.0012 and β=0.001 (water jet in air). The limit model may fail to describe low-Re situations, where the influence of the outer fluid cannot be ignored.

Steady micro-jetting may give rise to greater productivity (larger liquid flow rates), with a well controlled and small drop size. On the contrary, dripping may give rise to much larger, isolated droplets under similar Re and We. Dripping usually yields highly monodisperse spray, but it may also exhibit bi-disperse or polydisperse droplet distributions. Here, such non-linear complex dynamics may arise whenever a given parametrical instance {Re,We,α,β} leads to more than one absolutely unstable wave solutions and these solutions exhibit similar local growth rate ω_(i) but different local oscillation frequency ω_(r).

Another analysis of the dispersion model comprises the solution for each combination of {Re,We,α,β} with the largest temporal growth rate Im(ω)>0, i.e. the one that dominates the break-up dynamics. FIG. 9 c depicts a graph of absolute/convective instability transition loci in the {Re, We} plane for different values of {α,β}. Regions below (above) the curves correspond to absolutely (convectively) unstable configurations. Thus, the graph depicts the loci in the {Re, We} plane for which Im(ω) changes its sign. The line formed by these points mark the transition from a convective to an absolute (C/A) character of the instability for a given set of values of {α,β}.

Circles in FIG. 9 show the critical We numbers found with the numerical code for α=β=1000 (n_(i)=31 and n_(e)=71 collocations points were used). There is a close agreement with the results obtained by solving the analytical dispersion relation (2). From the results above, one obtains that in the limit Re <<1, for given {α,⊖}, the C/A transition depends on the capillary number Ca=We/Re only. For example, for α=β, this results in an expression for the critical capillary number as Ca*=0.139α^(1/2) for α above 100. Below this critical capillary number, the instability is absolute (i.e., a gas spout breaks up—or bubbles—locally).

One result of the analysis is that in the limit of vanishing viscosity and density of the inner fluid (α,β→∞), the jet may be absolutely unstable for any combination of values of the Reynolds and Weber numbers. When the viscosity and density of the inner fluid are small but finite, for any value of the Reynolds number (either low or high) there may be a finite value of the Weber number at which the transition from an absolute to a convective character of the instability takes place.

The dispersion model may be generated analytically, such as by using the formulas shown above. Or, it may be generated numerically, such as by analyzing experimental data. The dispersion model developed analytically was compared with experimental results. A stainless steel fluid focusing device with dimensions D=D₁=150 μm, L=160 μm, and H=125 μm is used. The inner edge of the orifice has been rounded to minimize vena contracta effects. The ambient focusing fluid is distilled water at T=23° C. (ρ₂=995 kg·m⁻¹, μ₂=0.001 Pa·s). Three silicone oils with nominal Newtonian viscosities μ₁ equal to 0.005, 0.02 and 0.1 Pa·s and measured oil-water surface tensions ρ=33.2, 30.4, and 28 mN/m at T=23° C. are used as the jet-forming liquid. The oil densities are ρ₁=965±0.2% kg·m⁻¹. Surfactants need not be used. Thus, the study explores three values β=0.2, 0.05, and 0.01, with α=1.033. FIG. 10 depicts the theoretical jetting-dripping transition for various (α,β,Re,We). The dotted lines show the experiments in FIGS. 12 a-12 f (small, joined symbols: jetting; isolated large symbols: dripping); Q₂=5.16 mL/min (empty symbols) and 1.82 mL/min (full symbols)

To compare the experiments with the theory, the diameter of the unperturbed jet radius at the orifice exit is estimated by assuming both liquids to issue at an approximately uniform velocity V=4(Q₁+Q₂)/(πD²), Q₁ and Q₂ being the oil and water flow rates (Re₂ ranges from 262 to 1362): R_(j)˜(D/2)(Q₁/Q₁+Q₂))^(1/2). Knowing R_(j) and Re, We can be calculated. There are dispersed five oil flow rates Q₁=2, 5, 10, 25, 50, and 100 ml/h, using a syringe pump Harvard Apparatus mod. 4455 with B-D 1 cc plastic syringes (Q₁=1 and 2 ml/h), 5 cc (Q₁=10 ml/h) and 20 cc (Q₁=25 and 50 ml/h). Two water flow rates Q₂=1.82 and 5.16 ml/min (±0.5%) were supplied, using a pressurized 300 cc water reservoir.

FIGS. 12 a-f summarize one set of experimental results. Experiments are depicted for diverse values of Q₁, Q₂ and μ₁ (α=1.033; β=0.01 in FIG. 12 a and FIG. 12 b; β=0.05 in FIG. 12 c and FIG. 12 d; β=0.2 in FIG. 12 e and FIG. 12 f; exposure time: 100 ns. The scale is given by the dash, measuring 200 μm. FIG. 12 e contains an insert showing the jet breakup region. The jetting-dripping transition in flow focusing exhibits a significant hysteresis effect: i.e., parametrical ranges where (depending on the route followed) meta-stable dripping or jetting may occur in regions corresponding to jetting or dripping, respectively. Hysteresis may be caused by the global stability of the system consisting of the capillary feeding tube, the funnel-like meniscus attached to it, and the issuing jet. When the steady jetting regime is abandoned to enter the dripping regime, a considerable part of the apex region (at times, the whole meniscus) oscillates, thus precluding the immediate recovery of steady jet flow conditions, even after the operational parameters return to their original setup in the former jetting regime. To enforce the recovery of jetting conditions, the system may be turned back sufficiently far into the convectively unstable (jetting) parametrical subspace. The amplitude of the hysteretic effect is associated to the meniscus size, i.e., to the relation between feed tube size and orifice diameter: the smaller the feed tube, the smaller the jetting-dripping hysteresis, and the reverse. This is analogous to the jetting-dripping transition in electrospray. Moreover, the higher the liquid viscosity, the larger the associated hysteresis effect. In the experiments, a sharp-edged feed tube (FIG. 2) was chosen whose inner diameter D₁ coincides with the orifice diameter. The hysteresis effect was thus minimized while maintaining a genuine fluid focusing geometry. FIGS. 12 a-f shows whenever it is feasible, the hysteresis range of each experimental sequence.

Therefore, the experiments depicted in FIGS. 10 and 12 a-f go from jetting to dripping and not the other way around, because the re-stabilization of a dripping mode requires driving the system considerably deep inside the jetting mode. In FIGS. 12 a-f, owing to setup limitations, only four of the six experimental sequences were brought to the dripping regime.

FIGS. 9 b and 10 show a surprising jetting-dripping transition topology: the upper branch is slanted so that a potential for double transition exists; should one move along a constant-Re line, one would first hit the jetting-dripping transition and subsequently cross a jetting-dripping transition. Of course, such result is to be taken cautiously, because of hysteresis. The potential for double transition may be theoretical, since experiments do not follow a constant-Reynolds route: when one gradually increase the axial speed V, both the Weber and Reynolds numbers grow (see Eq. 1), so that one would move over a parabolic line crossing the jetting-dripping border just once. Another intrinsic experimental difficulty is the water ambient, whose divergent flow after the orifice exit tends to open up the core stream leading to radial dispersal of the drops. The choice of a sufficiently large L distance (L=160 μm>D) helps to preserve parallel flow conditions during a distance long enough to give rise to a roughly cylindrical oil jet.

In spite of experimental difficulties, the model provides an acceptable fit of the jetting-dripping transition data as illustrated by the pictures in FIGS. 12 a-f. Experimental dot curves in FIG. 10 show a sequence of five measurements with different core liquid flow rates Q₁; the remaining physical parameters are fixed. The jetting-dripping transition takes place at the lowest Q₁ flow rate compatible with a steady jet. Any flow rate below such threshold produces the emission of large, isolated drops (dripping): the drop-producing mechanism is the sequential filling and detachment from a pulsating axial filament, an elongated version of faucet-dripping (see FIG. 12 d). Jetting drops may show a diameter roughly similar to the jet diameter; while dripping drops are substantially larger. The jetting-dripping transition is remarkably predicted for the smallest viscosity oil (FIG. 12 a) and Q₂=1.82 ml/min: it takes place at Q₁=4.9 ml/h. The transition is also very well predicted at β=0.05 for both Q₂ flow rates. When dealing with the largest viscosity oil (FIGS. 12 e-f), the transition may be more difficult to assess, although the experimental results fall very close to the predicted transition within the experimental uncertainties associated to the hysteresis effect. In FIGS. 12 e-f, the top single pictures show jetting behavior, while the remaining shots illustrate a typical dripping process: a long pulsating tendril is filling a thick drop; some recoil is observed, owing to braking effects from the ambient liquid.

Additional comparison of results is shown in FIG. 13 which shows bubbling to jetting. Liquid: ethanol-water (50/50% vv), T=28° C. Orifice diameter D=200 μm. (a) Q₁=13 mL/min, Q_(g)=0.65 mL/min; (b) Q₁=23 mL/min, Q_(g)=0.25 m/min; (c) Q₁=23 mL/min, Q_(g)=0.5 mL/min. The arrow indicates the flow direction. The results are of experiments where a gaseous jet was pulled by a stream of liquid flowing through a co-axial orifice of diameter D (Flow Focusing technique). Both the liquid and the gas issued from a stagnant region upstream of the orifice, where pressure is kept constant and equal for both fluids. In order to assess the diameter of the undisturbed gaseous jet, the assumption was made that the gas is withdrawn by tangential viscous stresses by the liquid stream, and therefore that the gas velocity inside the undisturbed gas jet diameter does not depart significantly from that of the co-flowing liquid. Under this assumption, mass conservation for an incompressible fluid yields: d_(j)=[4Q_(g)/(πU)]^(0.5). This approximation may be more accurate for smaller radii of the gaseous jets (owing to the growing importance of viscous action inside them) and for smaller driving pressure gradients. An analysis of the equations of the steady jet (in the high Re regime) may lead to the conclusion that a plug-flow velocity profile in the jet at the orifice exit is a sufficiently accurate solution, and the only one consistent with the use of the dispersion relation. In the experiments, it has been observed what seems a clear transition to jetting, as shown in FIG. 13, although the onset of convective instability is often blurred by unsteadiness and chaotic bubbling. FIG. 14 plots the experimental data for three representative liquids (corresponding to the non dimensional parameters {α=760,β=68}, {α=800,β=324}, and {α=850,β=544}), as well as the experiments of FIG. 13 (under the assumption that R_(o)<<D). All experimental data points, which correspond to monodisperse microbubbling, are located below the calculated Weber numbers corresponding to the transition from bubbling to jetting. The observation that all experimental data for bubbling fall into the region of absolute instability established by the model supports it.

One conclusion from this analysis is that a cylindrical gas spout in a high viscosity liquid moving with speed Vmay be absolutely unstable, and may prevent the formation of long gas spouts, below a finite value of the Weber number or, alternatively, below a finite value of the Capillary number Ca=We/Re. Thus, the solution for gas spouts entrained in extensional viscous flows discussed in regard to FIG. 11, may be convectively unstable as long as the outer liquid is sufficiently viscous, provided that there is minimum finite Capillary number for which the steady solution can exist. For Re<<1, below the critical value of the Capillary number, the spouts formed by the extensional flow in the solution may form a train of microbubbles, a fact observed in related experiments.

In those experiments, to provide a flow as close as possible to the true extensional one, the liquid stream is surrounded by another gas stream in order to avoid the liquid contact with the exit orifice wall. This is not a true unbounded liquid flow, but the diameter of the spout is small in comparison to the outer diameter of the liquid-jet, and the results illustrate well our findings. The influence on the ratio d_(l)/d_(g) of the outer diameter of the liquid-jet to the diameter of the gaseous jet is verified using an alternative spectral collocation code. For d_(l)/d_(g)=100, the differences in the critical Weber numbers with those of unbounded liquid were less than 3% for large Reynolds numbers, while the differences were less than 1% for small Reynolds numbers.

The model may be applied in a variety of instances. For example, a hollow optical fiber with a cylindrical hole may only be drawn if the Weber number is above the critical one, or if there is an alternative mechanism providing local stability. This mechanism may be a longitudinal positive gradient of viscosity (e.g., the one due to “fiber quenching”, when the glass solidifies) or an external negative pressure gradient (a potential alternative in ultra-high speed fiber drawing).

The model may also be applied in other configurations, such as that shown in FIGS. 15-17. FIG. 15 depicts the flow leading to (after solidification downstream) creation of a single fiber containing multiple interior cylinders which are either hollow, if a gas is used as the inner fluid, or are composed of a second material. As shown in FIG. 15, there are three nozzles that extend a certain finite distance, and each of which axisymmetrically may constrain a fluid of one composition (“Composition B”) around a second entrained fluid of a different composition (“Composition A”), and which themselves may be surrounded by a larger diameter flow of more fluid of Composition B. This may be an efficient way to manufacture, for example, special optical fibers. Second, nozzles may be aligned along a line, allowing the fluid inside the nozzle to be entrained by a two-dimensional viscous flow in a thin liquid film. For example, FIG. 16 depicts nozzles aligned along a line, allowing the fluid inside the nozzle to be entrained by a two-dimensional viscous flow in a thin liquid film. After solidification this creates a thin sheet of material with thin ribbons which are either hollow or comprised of a second material. An example of this is depicted in FIG. 17, which illustrates a sheet 1700 containing parallel tubes 1702 prepared by aligning nozzles along a line. The tubes 1702 may be hollow or filled with a material different from that of the bulk. The figures, including FIG. 17, are not necessarily to scale. Such composite thin sheets with precisely patterned texture may have interesting material properties. For example, they may be very strong in one direction (perpendicular to the ribbons) and more fragile in the direction along the ribbons. An arrangement of parallel ribbons containing parallel long holes 1802 along them can be packed in the form of a “long brick” 1800 with many parallel long holes along their length distributed as shown in FIG. 18.

This methodology enables creating compound fibers wherein the inner diameter can be made far smaller than the outer diameter of the fiber. The disparity in the interior and exterior diameters has at least two useful consequences. First, by placing a group of nozzles in close proximity, the entrainment process can be used to create a single fiber containing multiple interior cylinders which are either hollow or made of a second material (FIG. 16).

It is therefore intended that the foregoing detailed description be regarded as illustrative rather than limiting, and that it be understood that it is the following claims, including all equivalents, that are intended to define the spirit and scope of this invention. Other variations may be readily substituted and combined to achieve particular design goals or accommodate particular materials or manufacturing processes. 

1. A filament being produced by: entraining a first fluid within a second fluid by flowing a third fluid, the flowing third fluid at least partly constraining the second fluid in at least one dimension.
 2. The filament of claim 1, further comprising solidifying at least a part of the second fluid to form a fiber.
 3. The filament of claim 2, further comprising solidifying at least a part of the first fluid to form the fiber.
 4. The filament of claim 1, wherein entraining a first fluid within a second fluid is based only on fluid properties of the first fluid and the second fluid
 5. The filament of claim 4, wherein the fluid properties are viscosity of the first fluid and the second fluid; and wherein the second fluid is more viscous than the first fluid.
 6. The filament of claim 1, wherein the first fluid comprises a liquid or a gas that is immiscible with the second fluid.
 7. The filament of claim 1, wherein a plurality of fluids are entrained within the second fluid.
 8. The filament of claim 7, wherein the third fluid flows at least partly by co-electrospinning.
 9. The filament of claim 7, wherein the third fluid flows at least partly by pressure applied to the third fluid.
 10. The filament of claim 7, wherein the second fluid is axisymmetrically constrained by the flowing of the third fluid to surround the second fluid.
 11. The filament of claim 7, wherein the first fluid flows from a first nozzle and the second fluid flows from a second nozzle; and wherein the first nozzle is concentric with the second nozzle.
 12. The filament of claim 7, wherein the second fluid flows to entrain the first fluid; and wherein the flow of the second fluid is due more to drawing of the second fluid by the flowing of the third fluid than to pressure applied to the second fluid.
 13. The filament of claim 12, wherein the entraining of the first fluid within the second fluid is due more to drawing of the first fluid by the flowing of the second fluid than to pressure applied to the first fluid.
 14. A filament being produced by: entraining a first fluid within a second fluid based on a model of a dynamic response of the first and second fluids as functions of densities of the first and second fluids, viscosities of the first and second fluids, Reynolds numbers of the first and second fluids, and Weber numbers of the first and second fluids.
 15. The filament of claim 14, wherein the model of the dynamic response defines a phase space of a stable entraining of the first fluid within the second fluid; and wherein values of the Reynolds numbers of the first and second fluid and Weber numbers of the first and second fluid are selected to be in the phase space of the stable entraining of the first fluid within the second fluid.
 16. The filament of claim 15, wherein the entraining of the first fluid within the second fluid is performed by a system that controls velocity of the first fluid and second fluid; and wherein the velocity of the first fluid and the second fluid are selected such that the model is in the phase space of the stable entraining of the first fluid within the second fluid.
 17. The filament of claim 15, wherein the model is derived numerically.
 18. The filament of claim 15, wherein the model is derived analytically.
 19. The filament of claim 18, wherein the model is a function of: the ratio of the densities of the first and second fluid, the ratio of the viscosities of the first and second fluid, the Reynolds number of the first fluid Re ₁=ρ₁ VR ₁/μ₁ with ρ₁ being the density of the first fluid, μ₁ being the viscosity of the first fluid, R₁ being the radius of a cylindrical jet through which the first fluid flows, and Vbeing the uniform velocity of the first and second fluids relative to an observer, the Reynolds number of the second fluid Re ₂=ρ₂ VR ₂/μ₂ with ρ₂ being the density of the first fluid, μ₂ being the viscosity of the first fluid, and R₂ being the radius of a cylindrical jet through which the second fluid flows, the Weber number of the first fluid We ₁=ρ₁ V ² R ₁/σ where σ being the surface tension, and the Weber number of the second fluid We ₂=ρ₂ V ² R ₂/σ. 20 The filament of claim 14, further comprising solidifying at least a part of the second fluid to form a fiber.
 21. The filament of claim 14, wherein operating conditions are selected in order to be in a specific phase space within the model so that the flow of the first fluid is continuous for a predetermined time.
 22. The filament of claim 21, wherein the operating condition comprises pressure; and wherein the pressure to at least one of the first and second fluids is dynamically adjusted to maintain a shape of an interface between the first fluid and the second fluid for the predetermined time. 